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## Cloud Guide

piecewise_cs.cs

### piecewise_cs.cs

```/* Copyright 2018, Gurobi Optimization, LLC */

/* This example considers the following separable, convex problem:

minimize    f(x) - y + g(z)
subject to  x + 2 y + 3 z <= 4
x +   y       >= 1
x,    y,    z <= 1

where f(u) = exp(-u) and g(u) = 2 u^2 - 4 u, for all real u. It
formulates and solves a simpler LP model by approximating f and
g with piecewise-linear functions. Then it transforms the model
into a MIP by negating the approximation for f, which corresponds
to a non-convex piecewise-linear function, and solves it again.
*/

using System;
using Gurobi;

class piecewise_cs
{

private static double f(double u) { return Math.Exp(-u); }
private static double g(double u) { return 2 * u * u - 4 * u; }

static void Main()
{
try {

// Create environment

GRBEnv env = new GRBEnv();

// Create a new model

GRBModel model = new GRBModel(env);

// Create variables

double lb = 0.0, ub = 1.0;

GRBVar x = model.AddVar(lb, ub, 0.0, GRB.CONTINUOUS, "x");
GRBVar y = model.AddVar(lb, ub, 0.0, GRB.CONTINUOUS, "y");
GRBVar z = model.AddVar(lb, ub, 0.0, GRB.CONTINUOUS, "z");

// Set objective for y

model.SetObjective(-y);

// Add piecewise-linear objective functions for x and z

int npts = 101;
double[] ptu = new double[npts];
double[] ptf = new double[npts];
double[] ptg = new double[npts];

for (int i = 0; i < npts; i++) {
ptu[i] = lb + (ub - lb) * i / (npts - 1);
ptf[i] = f(ptu[i]);
ptg[i] = g(ptu[i]);
}

model.SetPWLObj(x, ptu, ptf);
model.SetPWLObj(z, ptu, ptg);

// Add constraint: x + 2 y + 3 z <= 4

model.AddConstr(x + 2 * y + 3 * z <= 4.0, "c0");

// Add constraint: x + y >= 1

model.AddConstr(x + y >= 1.0, "c1");

// Optimize model as an LP

model.Optimize();

Console.WriteLine("IsMIP: " + model.IsMIP);

Console.WriteLine(x.VarName + " " + x.X);
Console.WriteLine(y.VarName + " " + y.X);
Console.WriteLine(z.VarName + " " + z.X);

Console.WriteLine("Obj: " + model.ObjVal);

Console.WriteLine();

// Negate piecewise-linear objective function for x

for (int i = 0; i < npts; i++) {
ptf[i] = -ptf[i];
}

model.SetPWLObj(x, ptu, ptf);

// Optimize model as a MIP

model.Optimize();

Console.WriteLine("IsMIP: " + model.IsMIP);

Console.WriteLine(x.VarName + " " + x.X);
Console.WriteLine(y.VarName + " " + y.X);
Console.WriteLine(z.VarName + " " + z.X);

Console.WriteLine("Obj: " + model.ObjVal);

// Dispose of model and environment

model.Dispose();
env.Dispose();

} catch (GRBException e) {
Console.WriteLine("Error code: " + e.ErrorCode + ". " + e.Message);
}
}
}
```